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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Unknotted gropes, Whitney towers, and doubly slicing knots


Authors: Jae Choon Cha and Taehee Kim
Journal: Trans. Amer. Math. Soc. 371 (2019), 2383-2429
MSC (2010): Primary 57N13; Secondary 57N70
DOI: https://doi.org/10.1090/tran/7371
Published electronically: October 26, 2018
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Abstract: We study the structure of the exteriors of gropes and Whitney towers in dimension 4, focusing on their fundamental groups. In particular we introduce a notion of unknottedness of gropes and Whitney towers in the 4-sphere. We prove that various modifications of gropes and Whitney towers preserve the unknottedness and do not enlarge the fundamental group. We exhibit handlebody structures of the exteriors of gropes and Whitney towers constructed by earlier methods of Cochran, Teichner, Horn, and the first author and use them to construct examples of unknotted gropes and Whitney towers. As an application, we introduce geometric bi-filtrations of knots which approximate the double sliceness in terms of unknotted gropes and Whitney towers. We prove that the bi-filtrations do not stabilize at any stage.


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Additional Information

Jae Choon Cha
Affiliation: Department of Mathematics, POSTECH, Pohang 37673, Republic of Korea – and – School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
Email: jccha@postech.ac.kr

Taehee Kim
Affiliation: Department of Mathematics, Konkuk University, Seoul 05029, Republic of Korea
Email: tkim@konkuk.ac.kr

DOI: https://doi.org/10.1090/tran/7371
Received by editor(s): January 24, 2017
Received by editor(s) in revised form: August 10, 2017
Published electronically: October 26, 2018
Additional Notes: The first named author was supported by NRF grant 2011-0030044
The second named author was supported by NRF grant 2011-0030044 and NRF grant 2015R1D1A1A01056634
Article copyright: © Copyright 2018 American Mathematical Society

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